The paper describes an algorithm for numerical estimation of effective mechanical properties in twodimensional case, considering finite strains. The algorithm is based on consecutive application of different boundary conditions to representative surface elements (RSEs) in terms of displacements, solution of elastic boundary value problem for each case, and averaging the stress field obtained. Effective properties are estimated as a quadratic dependence of the second PiolaKirchhoff stress tensor upon the Green strain tensor. The results of numerical estimation of effective mechanical properties of plexiglas, reinforced with steel wire, are presented at finite strains. Numerical calculations were performed with the help of CAE Fidesys using the finite element method. The dependence of the effective properties of reinforced plexiglas upon the concentration of wires and the shape of wire cross section is investigated. In particular, it was found that the aspect ratio of reinforcing wire cross section has the most significant impact on effective moduli characterizing the material properties in the direction of larger side of the cross section. The obtained results allow one to estimate the influence of nonlinear effects upon the mechanical properties of the composite.
The widespread use of plexiglas in engineering applications is due to its properties such as transparency, strength, flexibility, lightness, cheapness, and nontoxicity. In some applications the combination of plexiglas properties of high rigidity and resistance to mechanical and thermal effects is required. In order to increase stiffness and thermal conductivity of the plexiglas and to prevent its spillage due to mechanical or other influences a reinforcement wirework (usually made of steel) is used. Reinforced plexiglas is a composite material. Properties of the material obtained depend primarily upon the material and shape of reinforcing wire. At that, the question that has to be answered is as follows: how to evaluate mechanical properties of reinforced plexiglas while mechanical properties of plexiglas and wires are known provided the shape of wire cross section is known?
The averaging of heterogeneous materials properties has been of interest since the middle of the last century. Theoretical principles of such averaging are described in [
For composites with relatively small volumetric content of filler in the matrix, the existing bilateral assessment of Hashin and Shtrikman [
At present, the more topical problem is evaluation of effective mechanical properties of heterogeneous materials in nonlinear form, with the help of which it is possible to describe the behavior of composites under finite strains. In the work [
This work proposes an algorithm for construction of effective constitutive relations of composite materials in nonlinear elastic form under finite strains. An evaluation of the effective characteristics is discussed for twodimensional case, which is a simplification as compared to the threedimensional case for which the algorithm is described in [
The paper presents the results of twodimensional calculations of reinforced plexiglas effective characteristics. Calculations are performed with the help of CAE Fidesys using the finite element method. Plexiglas and its reinforcing steel wire were modeled by the Murnaghan material. A dependence of the effective elastic moduli on the concentration of the reinforcing wire and the shape of its cross section was investigated.
Let us present the basic designations and relations of the nonlinear theory of elasticity [
Let us call
Using the above definitions and designation, let us describe the algorithm for estimating effective characteristic of composite material for twodimensional case in the nonlinear form under finite strains.
For RSE
Each sequence of problems corresponds to a certain type of boundary conditions (
Solving each problem of each sequence we will find the field of the true stress tensor
The second equality in (
Knowing the deformation gradient that was specified in (
In a linear case, the effective constitutive relations are estimated as a linear dependence of true stress tensor
In the nonlinear case, the obtained effective true stress tensor
In this case, the effective properties are estimated as quadratic dependence of the second PiolaKirchhoff stress tensor
Thus, estimation of the effective properties of the composite in linear case reduces to the calculation of coefficients
At that, the following conditions of symmetry are in force for tensor components
The same conditions are in force for
Considering the above conditions of symmetry, there are 21 independent constants
Implementation of the algorithm relies on the use of finite element method [
The algorithm consists of the following logical blocks.
After that, the model is shifted to put its center to the origin (i.e., the coordinates of all mesh nodes are decreased by
In addition, the accuracy
The value of
If the problem is solved for
If the problem is solved for
If the condition with number
Then, a pair of corresponding nodes (i.e., those, the projection of which is closest to each other) are formed from opposite edges of the RSE. The pairs are recorded to two lists.
For this purpose, a pass through all nodes of the edge
Similarly, a pass through all nodes of the edge
where
There are 6 sequences, each of which contains 34 problems or more, if required. Different problems within the same sequence differ in the amount of strain
On the basis of effective strain tensor, the effective deformation gradient
Formulas for gradient’s components in an explicit form are as follows:
For
For
For each pair of nodes from the second list (
First, the area of the model in the final state (i.e., after deformation) is calculated using the formula:
In other words, the area of model in final state represents the area in initial state multiplied by the determinant of effective deformation gradient.
Stress tensor is averaged over the area using formula (
Hereby one can calculate the effective true stresses tensor in the RSE. Knowing it, it is possible to calculate the effective second PiolaKirchhoff stress tensor using formula (
Blocks 4 and 5 of the algorithm should be run in double loop: by type of strain (1 to 6) and by amount of strain (from 1 to 3…4 or more). The result is as follows: for each boundary value problem for each sequence an effective Green strain tensor was set, and as a result the effective second PiolaKirchhoff stress tensor was obtained. The resulting PiolaKirchhoff tensors are stored for each problem for later calculation of effective properties in the form of their relations.
The dependence is constructed using the least squares method, which allows calculating coefficients
The relationship between the calculated coefficients
Formulas for calculation of coefficients
For calculation of coefficients
This system is solved analytically, and the solution is as follows:
Thus, the described algorithm allows performing numerical (using the finite element analysis) estimation of effective elastic moduli (of the first and the second order) for the composite material in the twodimensional case. On the basis of the described algorithm we have developed a software module Fidesys Composite as part of CAE system Fidesys [
The proposed algorithm assumes that the computations for each geometrical pattern are performed separately. For composites of periodical structure it is sufficient to perform computations once. For composites of random structure with statistically uniform distribution of inclusions one can use the approach that is based on the ensemble averaging over a fixed number of configurations [
With the help of the developed module Fidesys Composite, two series of finite element analyses of effective characteristics of plexiglas reinforced with steel wire were conducted for twodimensional case in the nonlinear form under finite strains. A rectangular RSE of plexiglas with a rectangle of steel modeling wire was considered in all calculations. A level of strains applied to the RSE was 1%, 2%, and 3%. Mechanical properties of plexiglas and steel were described using the Murnaghan constitutive relations [
The constants
Dependence of effective properties of reinforced plexiglas versus the concentration of the reinforcing wires was studied. RSE was a rectangle 10 × 5 mm. Cross section of wire is square; size of square side ranged from 0.25 mm to 3 mm.
Let us see graphs for linear coefficients
Dependence of coefficient
Dependence of coefficient
Dependence of coefficient
Dependence of coefficient
The graphs show that the coefficient
Let us see graphs for nonlinear coefficients
Dependence of coefficient
Dependence of coefficient
Dependence of coefficient
As can be seen from the graphs, the coefficient
Dependence of effective properties of reinforced plexiglas versus the form of cross section of the reinforcing wires was studied. RSE was a square 10 × 10 mm. The rectangular form of wire cross section was considered, the ratio of rectangle sides varied from 1 : 1 to 1 : 10. At that, the crosssection area of the wire was 9 mm^{2}.
Let us see graphs for linear coefficients
Dependence of coefficient
Dependence of coefficient
The graphs show that the coefficient
Let us see graphs for nonlinear coefficients
Dependence of coefficient
Dependence of coefficient
Dependence of coefficient
As can be seen from the graphs, the coefficient
The points of inflection in the figures showing stiffness constants as a function of the width/length ratio of the inclusions arise apparently due to computational errors.
Within the framework of linear elasticity and small strains, the obtained results are compared with the results of Christensen [
Linear effective elastic modulus of the fiberreinforced composite. A comparison between the Fidesys computations and the results of Christensen [
Effective moduli, MPa 





Fidesys  3.13691  1.30607  0.89212  3.13692 
Christensen [ 
3.11029  1.33286  0.88872  3.11029 
One can see from Table
Another comparison was performed for the case of finite strains. The pure shear of a fiberreinforced elastomeric composite with rigid circular fibers (wires) was considered. The matrix material was incompressible, and the mechanical properties of the matrix material were described by the neoHookean potential. The stresses were averaged over the RSE, and the first Piola shear stress was computed. The results were compared with the estimations of Avazmohammadi and Ponte Castañeda [
The first PiolaKirchhoff shear stress in a fibrous elastomeric composite as a function of shear strain. A comparison between the Fidesys computations and the results of Avazmohammadi and Ponte Castañeda [





Fidesys  Avazmohammadi and Ponte Castañeda  Fidesys  Avazmohammadi and Ponte Castañeda  
1.25  0.83  0.94  1.34  1.12 
1.50  1.93  1.48  2.81  1.92 
Thus, the paper presents the algorithm for numerical evaluation of the effective mechanical properties of nonlinear elastic solids in twodimensional case (under plane strain). The novelty of this algorithm is determined by considering of nonlinear effects. Both physical and geometric nonlinearity is taken into account. Effective constitutive relations are presented in the form of quadratic dependence of averaged strains versus stresses. Determination of effective modules is reduced to solution of sequences of nonlinear boundary value problems of elasticity for loads of different types and sizes. For solving boundary value problems the finite element method is used, it is implemented in CAE system Fidesys. Results of calculation of reinforced plexiglas effective characteristics, presented in the paper, confirm the efficiency of the algorithm. The influence of concentration of reinforcing wires and wire shape upon the effective properties is studied. In particular, it was found that the aspect ratio of reinforcing wire cross section has the most significant impact on effective moduli characterizing the material properties in the direction of larger side of the cross section.
The obtained results allow estimating the influence of nonlinear effects upon the mechanical properties of the composite for different amounts of strain. For example, in Figures
In the future it is planned to perform similar calculations for other composite materials.
The authors declare no competing interests.
The research for this paper was performed in Fidesys LLC and was financially supported by Russian Ministry of Education and Science (Project no. 14.579.21.0076; Project ID RFMEFI57914X0076).